Characterization of eigenfunctions of the Laplacian on Riemannian symmetric spaces of noncompact type

A result of real analysis (John Roe 1980) states that if all derivatives and antiderivatives of a measurable function f are uniformly bounded then  f is a linear combination of $\sin x$ and $\cos x$. This result was extended to $\mathbb{R}^n$ independently by Howard and Reese (1992) and by Strichartz (1993) where the derivatives  and antiderivatives are substituted by integral powers of Laplacian on $\mathbb{R}^n$. While itis plausible to extend this theorem for other Riemannian manifolds, Strichartz showed that the result holds true for Heisenberg groups, but fails for hyperbolic spaces.  Hyperbolic spaces are the most distinguished prototypes of the rank one Riemannian symmetric spaces of noncompact type. Strichartz's counter example can be extended to this larger class. In this talk we shall try to find the root cause of this failure and explore the possibility of extending this theorem for this class of spaces.